Let `D-=[-1,1]` is the domain of the following functions state which of them are injective.

Options

A. `f(x)={(tan^(-1)1/x, x!=0),(1,x=0):}`
B. `g(x)=x^(3)`
C. `h(x)=sin2x`
D. `k(x)=sin(pix//2)`

Explanation

To determine which of the given functions are injective (one-to-one) over the domain \( D = [-1, 1] \), we will analyze each function step by step.

### Step 1: Understanding Injective Functions
An injective function is one where each element of the domain maps to a unique element in the codomain. This means that if \( f(a) = f(b) \), then \( a = b \). A graphical way to check for injectivity is to see if a horizontal line intersects the graph of the function more than once.

### Step 2: Analyze Each Function

1. **Function \( f(x) = \tan^{-1} \left( \frac{1}{x} \right) \)**

- The function \( f(x) \) is defined for \( x \in (-1, 1) \) except at \( x = 0 \).
- As \( x \) approaches \( -1 \), \( f(x) \) approaches \( -\frac{\pi}{4} \).
- As \( x \) approaches \( 1 \), \( f(x) \) approaches \( \frac{\pi}{4} \).
- The function is continuous and strictly increasing in the interval \( (-1, 1) \).
- **Conclusion**: This function is injective.

2. **Function \( g(x) = x^3 \)**

- The function \( g(x) \) is defined for all \( x \) and is a cubic function.
- It is strictly increasing for all real numbers.
- Therefore, for \( x \in [-1, 1] \), it will also be strictly increasing.
- **Conclusion**: This function is injective.

3. **Function \( h(x) = \sin(2x) \)**

- The function \( h(x) \) has a period of \( \pi \).
- Within the interval \( [-1, 1] \), the function will oscillate and take on the same values for different \( x \) (e.g., \( h(-1) = h(1) = 0 \)).
- **Conclusion**: This function is not injective.

4. **Function \( k(x) = \sin\left(\frac{\pi x}{2}\right) \)**

- The period of this function is \( 4 \).
- Within the interval \( [-1, 1] \), the function is strictly increasing.
- The values at the endpoints are \( k(-1) = -1 \) and \( k(1) = 1 \).
- **Conclusion**: This function is injective.

### Final Conclusion
The functions that are injective over the domain \( D = [-1, 1] \) are:

- \( f(x) = \tan^{-1} \left( \frac{1}{x} \right) \)
- \( g(x) = x^3 \)
- \( k(x) = \sin\left(\frac{\pi x}{2}\right) \)

The function \( h(x) = \sin(2x) \) is not injective.